The assumption on \(x\)  is necessary. If \(x\)  is algebraic of degree at most
\(n\)  and we take \(\Psi (q)=q^{-n}\)  (or
any other \(\Psi (q)=o(q^{-n+1})\) ), a variant of
Liouville's inequality [1]} implies that
\(B_{\Psi }^{\prime }(x)\)  consists essentially only
of the infinite set
of polynomial multiples of
its minimal polynomial not exceeding degree \(n\)  (including scalar multiples), up to finitely many exceptions.
Thus for any \(C>0\)  there are only
finitely many solutions to (REF ) in irreducible polynomials.
